There's no evidence the US had anything to do with the TSR.2's cancellation. It was entirely internal politics. Yes, the US wanted to push their own designs, but why not? They were in competition with the British industry, but also, on the other side of the coin, the US was keen on European nations being able to provide their own defence capabilities to counter the Soviet threat on mainland Europe.
A few facts. TSR.2 was built to GOR.343, not GOR.339. The latter was for the Canberra replacement spec, which produced two final designs, from English Electric (EE) and Supermarine (the last drawing board machine by the Supermarine team). The best aspects of these two designs were merged into what was eventually to become the TSR.2, but a fly in the ointment in the form of an agreement with Short Brothers by EE meant that that firm had to be written out of the equation, so GOR.343 was written to facilitate this and the winners of OR.339 were built as the TSR.2 under OR.343.
The actual reason behind the TSR.2's cancellation was its was just too expensive, but again, this wasn't necessarily the fault of the jet or the companies tasked with its design and construction. It was an entirely new concept in which the first prototype became the first pre-production aircraft; it's design was frozen before construction began. This was a new concept and it hampered the final design as the RAF Air Staff and the committee formed by the Ministry of Supply kept changing the goal posts as to what roles the TSR.2 was to actually carry out. The RAF saw it as a possible replacement for the V-bombers, too. OR.343 was re-written no less than four times to accommodate changes to the basic requirement that had moved beyond a simple Canberra replacement. No wonder it became too expensive. The negative press and lack of government (specifically MoD - the naval heads wanted the money after they had found out their big carrier projects were going to be scrapped) support also weighed on the project as a whole.
This lack of clarity with regards to the jet's ultimate role was directly as a result of Duncan Sandys' 1957 Defence White Paper, which cancelled a host of future weapon systems and concetrated on the development of rockets and missiles (during WW2 Sandys had been chair of a committee to investigate German V-1 and V-2 technology, where his love of rockets grew from). From this the Blue Streak Medium Range Ballistic Missile came about, but was cancelled in 1960 for the same reason as the TSR.2, cost, although rockets were supplied to the European Launcher Development Organisation as a first stage booster to the Europa rocket and launched continuously until 1971.
Cost affected the aviation industry as a whole and TSR.2 was not the only major aircraft design cancelled in early 1965. Simultaneously, the P-1154 supersonic Harrier, yes, it was going to be called the Harrier was cancelled, because the Navy and RAF couldn't decide on a standard that suited both requirements and the HS.681 four engined jet transport, built to support the TSR.2 in the field. Simultaneously, the British and French had entered a binding agreement to develop a supersonic airliner and this was seen as the future path for Britain's aerospace industry. How wrong they were, of course, but Concorde was not cheap. Something had to give and TSR.2 was it.
Regarding the USA, after the 1964 election, in which Labour under Harold Wilson defeated the Tories, the government sent aviation and defence representatives to the US to examine new projects over there, including a trip to Fort Worth, where Robert MacNamara's TF-X was being designed and built. He accompanied the British delegation. This gave the British an idea of what the US was up to and most likely planted the seed that brought about the cancellation of the three aircraft projects for US made products. Nevertheless, toward the end of 1964 the government assured the British Aircraft Corporation that TSR.2 and Concorde would not be cancelled and George Edwards, former Vickers design engineer and BAC chairman wrote a letter to BAC personnel (I have a copy of that letter in my stash) assuring them of the future of both these projects.
In the end, the government bought US products to replace its own, owing to the emphasis by the US designers that they would be cheaper and on paper the offer looked too good to refuse. The F.111 would replace the TSR.2, the F-4 Phantom II would replace the P-1154 and the C-130 Hercules would replace the HS.681.